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Strategies for Popular Number Puzzles

Strategy Families

There are two ways one can group all the strategies for Sudoku: by difficulty and by family. Difficulty is rather subjective but necessary, for example, when selecting the order in which strategies are tried by the solver. Some strategies will always be easier for some people to spot than others, but I believe I have chosen an ordering which is not too controversial. So the main documentation has a side menu organised by difficulty.

Below are these strategies organised into themes.

There is a new Introduction to Sudoku and an article on Brute Force vs Logic for those curious.

With chaining strategies, there is definitely a theme going through them. This theme is all about bi-value (only two candidates left in the same cell) and bi-location (only two occurrences of a particular candidate left in the same unit) pairs and the incredible number of deductions one can make from them. You will find, if you read through this group, that earlier strategies become part of a more general theory as the theme develops. Thus, for example, Remote Pairs are a subset of XY-Chains; that is, XY-Chains is a more general approach of which Remote Pairs are a specific instance. Do read the introductory articles Introducing Chains and Links and Weak and Strong Links.

Exotic strategies do overlap with chaining ones, but they have a peculiar flavour of their own and some wonderful, if obscure, logic. They are definitely worth presenting as a demonstration of people's ingenuity but you will only need to have recourse to them on the extreme puzzles.

There are naturally special strategies for Jigsaw and Killers because of their differences. These are now included for the first time on this site.

This strategy list is by no means complete. Many strategies can be further extended and we do not have a complete theory of all Sudoku puzzles. If you are interested in the concepts behind creation and grading, there is a PDF document here called Sudoku Creation and Grading. With the community's help I hope to extend the documentation here.

For those people wondering why "Escargot" cannot be solved by the solver, there is an article on this special Sudoku here. This is an early 'ultimate puzzle' but this crown has been usurped by the puzzle created by Arto Inkala, which is also in the example list.

I've been looking at a new idea for measuring the difficulty of very hard puzzles - ones that can't use the standard scoring because they don't complete.


Interesting Sudokus for other reasons.

Article: The Relative Incidence of Sudoku Strategies

Article: The 17 Clue 'Proof' - my take on the January 2012 computerised exhaustive search proof.

Article: The Chaos Within Sudoku - A Richter Scale on a paper by Maria Ercsey-Ravasz and Zoltan Toroczkai at the Faculty of Physics, Babes-Bolyai University, Romania. Very interesting paper.

Article: Generalizing Sudoku Strategies by Kevin Gromley, March 2014

My response to Crooks Algorithm is here.

I'm pleased to include on this web site the Sudoku Song (MP3 file) by Peter Levy (official web site here). Peter wrote and recorded this song a couple of years ago and managed to capture the essence of the Sudoku craze to great acclaim.

Making your own solver?

There are a couple of definition files that I use for calibrating and testing mine. These were created a long time ago and were very popular but people may have forgotten about them. Happy to host the files here:
  1. Ruud's top 50,000 Most very hard
  2. 31,513 17 clue puzzles. Hard to make these

    Continue to Introduction



    ... by: Hugh Pickering

    Sunday 25-Jun-2023
    Love your puzzles, but get stuck on the chains, or anything to complicated. However I am 81 years old and only started sudoku 18 months or so ago, so there maybe an excuse. You mention rules, but I see no reference to them in strategies. Perhaps you could direct me where I can find them. It might help me to progress to the next level.

    Andrew Stuart writes:
    Hi Hugh. Start Here. All the best
    Add to this Thread

    ... by: Anonymous

    Friday 17-Dec-2021
    It seems that when finding an APE pattern, the solver lists it as an Exocet and vice versa. Is this a bug?
    Andrew Stuart writes:
    Yes. Now fixed, as of a day or so ago
    Add to this Thread

    ... by: Anonymous

    Wednesday 8-Dec-2021
    Isn't Sue de Coq a Bent Set strategy?

    ... by: JAVIED AKBAR

    Saturday 16-Oct-2021
    Very informative excellent

    ... by: Wolftrakker

    Tuesday 17-Aug-2021
    Is there a way to not require all cages be given in the killer solver?
    Andrew Stuart writes:
    Add to this Thread

    ... by: Kim Uildriks

    Sunday 24-May-2020
    after all the obvious E2 = 9, F2 = 6, C1= 4

    ... by: Larry Craig

    Saturday 28-Dec-2019
    I have been doing several extreme sudoku puzzles a day for several months now. I have developed a strategy which helps me solve any puzzle, meaning, I don't need any other techniques. In the event it doesn't help me, like it didn't right now, that tells me that I am missing something simple, which I was. Obviously I haven't tried every puzzle. My only challenge now is to see if there is a puzzle it won't solve. Is there somewhere I can get extra extreme puzzles to check my theory further.
    Thank you

    ... by: Chucker

    Tuesday 1-Oct-2019
    Has this solver been written in C++?
    If C# or C++, is the source code available?
    Andrew Stuart writes:
    C mainly with a few C++ classes thrown in.
    Source code is propriety but I have orientated this site to programmers, there are enough information to code the strategies up. That's the fun part anyway.
    Add to this Thread

    ... by: Eric

    Monday 27-May-2019
    Please can you give me the link to your original step-by-step sudoku solver site? That was the one that give the exact strategy used to solve each cell.
    Andrew Stuart writes:
    Err, that has always been this solver
    Add to this Thread

    ... by: WW

    Saturday 10-Dec-2016
    What if you assumed the first candidate in the first unsolved cell was true? And now keep doing this(don`t eliminate as you go, I was thinking about blindly using trial and error) until you get a contradiction. The contradicting cell is now assumed to be the next candidate. If we run out of candidates in that cell to try, we change the cell before it to its next value. The idea was that by doing this we effectively search all possibilities(not really, but it doesn`t matter because if there is one contradiction the rest of the cell don`t matter) until we find the right one. But it might take a really long time. The worst-case scenario is that the correct combination is the last one, in which we have to sieve through n possibilities, where n is the product of the number of candidates in each unsolved cell. I am sure that it will work, but I want some confirmation.
    Andrew Stuart writes:
    This is exactly the algorithm the "Solution Count" button uses, with some tricks to make it more optimal. Brute force. If you wonder how many steps there is a count of that as well provided by the Solution Count. Note that for the same puzzle rotating it will produce a different result. Since its done from top left to buttom right Its about the clustering of numbers near the top left or away from it that effects the time. I think you can guess why. Solution Count will tell you the solution (or all solutions for a faulty puzzle) but it doesn’t tell you why the puzzle can be solved so its ultimately unsatisfying.

    Add to this Thread

    ... by: BLT

    Wednesday 27-Jul-2016
    You need a search window for beginners to look for rules and etc. while using the Suduko Solver and others. I read the answers but was dumbfounded by Rule 2 as I couldn't find it to refer to. How can the rules be referred to while working a puzzle?
    Andrew Stuart writes:
    Which strategy?
    All the results will highlight a strategy in the list to the right of the board. They are hyperlinked and should open in a new tab or window. There should be some documentation for everything returned by the solver so I'm interested to know where you were at this point
    Add to this Thread

    ... by: Jonathan

    Wednesday 30-Sep-2015
    You did a very good job collecting all these awsome strategies and presenting them here on this site and this really also has been great help for me solving sudokus.

    But I was wondering why you do not cover the 2-String-Kite, Skyscraper or Turbot Fish strategy, which i found on many other sudoku forums, as I think that they are extremly usefull for solving sudoku puzzles.

    ... by: Brooks DeTuncq

    Thursday 27-Nov-2014
    I am curious about strategies that don't seem to involve any of the basic or even the complex strategies listed on any of the Sudoku sites I have checked. I have been using all of the basic strategies listed on this site because I got to a point where combining rows and columns to eliminate candidates was no longer enough to complete the puzzles. I have yet to attempt the more complex strategies. I can work through the "Fiendish" puzzles listed in my App in 30 to 90 minutes. My daughter, who hasn't played as long as I have, can do the same puzzles in 7-10 minutes without marking any cells and not using any of the the basic or even complex strategies. She is unable to explain to me how she just sees the answers. Do you have any ideas how this would be possible? Is there anything you know of that discusses this type of phenomena?
    Andrew Stuart writes:
    The human brain is a wonderful thing. In addition to rational and logical thinking, it is capable of intuition. But intuition is not magical - it is using unconscious parts of the brain, especially those parts akin to pattern matching. I can't replicate that side of our solving ability, I can only mimic the logical strategies. In that sense this site is a shrine to formalism: strategies are proofs we only occasionally think of in those terms. I don’t know much about psychology, but I'm sure there are many studies in the area of intuition.
    Add to this Thread

    ... by: P Majumdar, Delhi, India.

    Thursday 20-Jun-2013
    I fully agree with the Comments from Mae (Thursday 9-Frb-2012.)

    All easy, medium and other low-difficulty level tough Sudoku puzzles cover only the very few basic simple strategies, whereas, tough/diabolic puzzles cover the remaining galaxy of advanced stategies. It is a pity!

    The creator/s of all tough/diabolic Sudoku puzzles, should, provide some minimum hint, at least by naming the strategies involved towards solving the puzzle.


    ... by: jennifer antonio

    Saturday 23-Jun-2012

    ... by: Mae

    Thursday 9-Feb-2012
    The so called easy Sudoku puzzles on Road Runner have just gotten too hard to make them fun any more.
    Sometimes it is fun to just take a few minute break and work and easy Sudoku. How can you encourage people to begin playing Sudoku, if you make them this tough?

    ... by: moses

    Wednesday 18-Jan-2012
    I am making my own booklet on sudoku strategies may i use your samples from your book?
    Andrew Stuart writes:
    Add to this Thread

    ... by: Jonathan Carr-Hopkins

    Wednesday 10-Nov-2010
    If, as Geoff in Australia offers, one cell can be only 3 or 5, then selecting one, and if that doesn't work, using the other, is this process, known as iteration, 'cheating', or is only a formulaic solution the only bona fide and acceptable one?

    ... by: Geoff

    Sunday 13-Jun-2010
    I have been doing the Sudokus out of the local paper (Queensland Times, Australia) and they are rated from One Star to Five Star.

    I have just run across the first Five Star Sudoku where I am 3/4 finished but can not find any more hints.

    Some of my friends tell me that you now need to start a 'trial & error' process ie if one cell can only be a 3 or a 5 - then pick on the 3 and see if it carries all the way through and if not, go back and pick on the 5.

    I tend to think that there should always be a 'hint' to be found - is this correct?


    ... by: Lloyd Welton

    Tuesday 18-May-2010
    Thanks for the great book. Never seen so many strategies so well explained in one book. Maybe in any following editions it should be spiral bound, that way it would a bit easier to use it as a reference.

    Thanks again.

    ... by: John C Raaen, Jr.

    Thursday 20-Aug-2009
    I blew it! The setup for a Virtual Wall is A1=1, A3=3, D2=4 and B4=4. Then A7, A9, C1, C2 and C3 are the "candidate squares for "4". Sorry!

    ... by: John C. Raaen, Jr

    Wednesday 19-Aug-2009
    I see no examples of my concept of a wall or virtual wall. If boxes are A thru I and the squares are 1 thru 9, fill A1=1, A2=2, A3=3. This is a wall. Any number other than 1, 2 or 3 in box B or C, say B1=4. can be placed only on squares C1, C2 or C3, and on A7, A8 or A9. Usually some of those candidate squares are already occupied, often leading to an easy placement of, in this case, a "4". (Of course, the squares B and C, 1, 2 and 3 are excluded in the above.)

    In a virtual wall, use the same setup but leave A2 blank and put the "2" on D2=2. Again, the "4" can be placed only on squares C1, C2 or C3 and on A7, A8 and A9.

    I am sure this simple technique is included somewhere on this site.
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